Questions tagged [bivariate-distributions]
For questions on bivariate distribution, the joint probability distribution of two random variables.
283
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Bivariate distribution of angles between two vectors and random vector
Let $Z_1 \in \mathbb{R}^d$ and $Z_2\in\mathbb{R}^d$ be two nonzero vectors. Let $X$ be a random vector distributed uniformly on the hypersphere $\mathbb{S}^{d-1}$. In the case $d=2$, the marginal ...
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Computing the probability using a joint density function
I am in the middle of an exercise dealing with two random variables $(X,Y)$ having the bivariate normal density function:
$$f_{(X,Y)}(x,y)=\frac{1}{2\pi\sqrt{1-\rho^{2}}}\exp {\left[-\frac{1}{2(1-\rho^...
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E[XY] where X and Y are the **sign functions** of standard normal distributions
The following question is from the book: "150 Most Frequently Asked Questions on Quant Interviews" By Stefanica, Radoicic, and Wang.
Let $X$ and $Y$ be standard normal variables with joint ...
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Find the CDF of Z when Z=XY for a given joint PDF for X and Y
The joint probability density function of random variables $ X$ and $ Y$ is given by
$$
f_{X,Y}(x, y) =
2(1-y) \quad \text{if} \quad 0 \leq x \leq 1, 0 \leq y \leq 1
$$
$$
f_{X,Y}(x, y) = 0 \quad \...
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Conditional mean and variance of two Gaussians given one is larger.
Given two Gaussian random variables $Z:=(Z_1,Z_2)\sim N(0,\Sigma)$, where $\Sigma\in\mathbb{R}^{2\times 2}$, is there an explicit expression for the mean and variance of $Z$ conditioned on $Z_1>Z_2$...
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Posterior distribution of a bivariate normal distribution
I have three continuous random variables $Z, X_1, X_2$ that are related as
$$ Z = a_1 \times X_1 + a_2 \times X_2 + \varepsilon = \textbf{a'} \textbf{X} + \varepsilon,$$
where $\varepsilon \sim N(0,1)$...
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Finding Correlation of a Bivariate Function
Here is a bivariate function with the variables $i,j$ is given as:
$f(i,j) = Wij(i+j),$
$0<i<1,$
$0<j<1.$
Where $W$ is just a constant. Assume that $M=\min(I,J)$ and $N = \max(I,J).$
...
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If there's another variable that controls whether heads outcomes are allowed, what distribution would that coin toss follow?
For a usual coin with probability of heads $P(H)$, the distribution of the numbers of heads after $n$ tosses follows a binomial distribution. The coin can be biased or unbiased.
But suppose there's ...
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Expectation in restricted bivariate distribution
I'm trying to follow the solution of this exercise: link.
It says: Let $X$ and $Y$ be two jointly continuous random variables with joint PDF
$
\begin{equation}
\nonumber f_{XY}(x,y) = \left\{
\begin{...
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bivariate normal distribution to standard normal distribution [duplicate]
I am not able to understand how it is comparing the given equation with bivariate normal dist. equation and finding the correct coefficients?
if i try to integrate and apply pdf properties ,it will ...
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Mean, Variance, and Conditional PMF from Poisson Process
In good years, quarrels between Tolstoy and his wife occurred according to a Poisson
process with rate λ = 5 per month. In bad years, it was a Poisson process with rate
μ = 10 per month. Suppose each ...
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How to derive conditional expectation E[X|Y=y] if X and Y follow bivariate normal distribution
I am struggling to derive $E[X|Y=y]=\mu_X+\sigma_X\rho(\frac{y-\mu_y}{\sigma_Y})$ when X and Y follow bivariate normal distribution.
I have read this, but I don't get how to get the following steps:
$$...
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Applying conditional independence
Let $X_1,\ldots,X_{n}$ be independent random variables, $n>2$ fixed and $\mathcal{F}_n$ the natural filtration. Consider an event $A\in \mathcal{F}_1$, and assume that given $A$ the laws of $X_2,\...
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Conditional distribution of a bivariate Gaussian/normal distribution
A bivariate Gaussian pdf is
$$
f_{X, Y}(x, y)=\frac{1}{2 \pi|\Sigma|^{1 / 2}} \exp \left(-\frac{1}{2}\left[x-m_1, y-m_2\right] \Sigma^{-1}\left[x-m_1, y-m_2\right]^T\right)
$$
where $\Sigma=\left[\...
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Why is the following variable a mixed random variable?
Let X∼N(0,1) and W∼Bernoulli(1/2) be independent random variables. Define the random variable Y as a function of X and W:
Find the PDF of Y and X+Y.
I don't understand why Z turned out to be a mixed ...
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Bivariate normal CFD approximation using characteristic function
The normal distribution CFD can be approximated using
$$F_X (x)=P[X≤x]=\frac{1}{2}-\frac{1}{π} \int^{\infty}_{0}\operatorname{Re}\left[\frac{e^{-iux}\phi_X (u)}{iu}\right]du$$
where the characteristic ...
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How to result in moment generating function of bivariate weibull distribution FGMBWD?
from here https://link.springer.com/article/10.1007/s40745-019-00197-5
The pdf of a FGMBW distribution is defined as
pdf
To prove the moment generating function start with
MGF
the result as follows
...
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Mutual information and functions of independent variables
Let $X, Y, Z$ be discrete random variables with $X$ and $Y$ independent of $Z$, while $X$ and $Y$ can be dependent. For the mutual information, we have $I(X; Y,Z) = I(X;Y)$. Now consider $I(X; f(Y,Z))$...
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The root of the sum of two normally distributed variables
Given $X,Y \sim \mathcal{N}(0,1)$ and $Z=\sqrt{X^2 + Y^2}$, find the PDF of $Z$.
I know from digging around that this will follow a Rayleigh distribution since the sum of two squared normally ...
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Log supermodularity property of Bivariate Normal CDF
I am trying to understand for bivariate normal variables, when their correlation increases, can we show in certian sense the distribution is concentrated on the 45 degree line.
For example, let $F(X,Y,...
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Probability of 2 dimensional random variable.
Two balls are selected at random without replacement from a box that contains 3 blue, 2 red, and 3 green balls. If $X$ is the number of blue balls selected and $Y$ is the number of red balls selected.
...
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finding constant in a bivariate join pdf
I'm given a
$$f_{X,Y}(x,y) =
\begin{cases}
cx, & \text{x > 0, y > 0, 1}\ \leq \ x+y \ \leq 2, \\
0, & \text{elsewhere.}
\end{cases}$$
and trying to find a the constant $c$.
I've set ...
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Derivative of the ration of bivariate normals wrt rho
Let $\tau_1 > \tau_2 > \tau_3$, and $\vec{x} \sim \mathcal{N}(\vec{0}, \Sigma_{\rho})$ with $\Sigma_\rho = \begin{pmatrix} 1 & \rho \\ \rho & 1 \end{pmatrix}$, that is $\vec{x}$ is a ...
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Marginal density equal to zero everywhere.
I've been working on a two part question on bivariate transformations and marginal densities but am having difficulty finding where I have made a mistake as the final answer for the marginal density ...
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Show that U and V are independent.
Suppose we have (X, Y ) be a bivariate normal vector with
mean vector µ = (0, 0) and covariance matrix
$$
Σ= \begin{bmatrix}2 & 1 \\1 & 2 \\\end{bmatrix}
$$
I'm given:
$$
A^TΣA=\begin{bmatrix}...
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The bivariate normal distribution and its ellipse
Let $f(x,y)$ be a bivariate normal p.d.f. and let $c$ be a positive constant so that $c < \left(2\pi \sigma_{X} \sigma_{Y}\sqrt{1-\rho^2}\right)^{-1}$. Show that $c=f(x,y)$ defines an ellipse in ...
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Using general bivariate gaussian to extract marginal PDF from given bivariate PDF
I had a homework question to find the marginal probability density functions, $p_X(x)$ and $p_Y(y)$, given a join probability density function $p_{XY}(x,y)$.
I have solved the problem by integration i....
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Distribution of a bivariate Gaussian
Let $X,Y$ be a bivariate normal distribution with correlation $\rho$, means $\mu_1 \mu_2$ and variances $\sigma_1^2, \sigma_2^2$. Is there an expression for $$P(X \ge 0, Y \ge 0)$$ that only uses $\...
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Confusing in limits of bivarite distribution and contradiction in related question
This question is from "Mathematical Statistics with application by Wackerly and Mendelhall" $8th$ edition page $233$ ,question $5.5$ :
$f(y_1,y_2)=3y_1 , 0 \leq y_2 \leq y_1 \leq1$ and $0$ ,...
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confusion on determining integral limits in bivarite distribution
$f(x_1,x_2)= 8x_1x_2 , 0<x_1<x_2<1 $ $\text{and}$ $0 , \text{elsewhere}$
I want to find $E(X_1X_2^2).$
According to the book: $$E(X_1X_2^2) = \int_{0}^{1}\int_{0}^{x_2} (x_1x_2^28x_1x_2)...
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Pdf of sum of independent rvs is the convolution of pdfs proof
I am trying to prove the statement in the title. However, I want some help to make my derivation mathematically rigorous.
I have $X_{1}$ and $X_{2}$ which are two independent rvs.
In addition, I have $...
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Covariance for a bivariate normal distribution
I have a question concerning the bivariate normal distribution where I have been able to prove that the first covariance we are asked for is equal to $0$, but was not sure if this would be implying ...
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Find $a$ such that the $P(X \in s) = 0.95$ where $ s = \{(x_1,x_2)^T \in \Bbb R : -a \leq x_1 \leq a \} $. Standard bivariate normal distribution.
I have a bivariate normal distribution, where $ X = (X_1, X_2)^T$.
The means are given by $\mu = (0,0)^T$ and the covariance matrix is $ \sum =
\begin{pmatrix}
1 & 0 \\
0 & 1 \\
\end{pmatrix}
...
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Prove Independence of a Bivariate Normal Distribution
Two variables $X,Y$ are bivariate normally distributed. We know that $Var(X)=Var(Y)$. Show that the two random variables $X+Y$ and $X-Y$ are independent.
I'm feeling pretty stumped by this question, ...
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Solution to bivariate normally distributed problem
Is anyone able to solve the following integral with a normal bivariate PDF, or at least to clarify if there exist a closed form solution?
Thanks in advance.
\begin{equation}
\frac{1}{2\pi} \int_{-\...
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Given probability density $f(x, y)=12xy(1-y)$, find the joint density of $U=XY^2$ and $V=Y$ and the marginals
Let $X$ and $Y$ be random variables with probability density $f(x, y)=12xy(1-y)$ if $0<x<1, 0<y<1$. Find the joint probability density of $U=XY^2$ and $V=Y$. Find the marginal density of $...
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Find marginal probability using bivariate normal
I'm given two variables $\begin{bmatrix} X_{1}\\{X_2} \end{bmatrix} \sim N_2({\mu}=\begin{bmatrix} 2\\-5\end{bmatrix},\Sigma = \begin{bmatrix} 1&{-0.5}\\{-0.5}&4 \end{bmatrix})$
How would I ...
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Expected Value of X,Y from Covariance Matrix
Let $(X,Y)$ be a bivariate random variable with a Gaussian distribution on $\mathbb{R}^2$, mean zero and variance-covariance matrix:
$$C=\begin{pmatrix} 0.42 & -0.42\\-0.42 & 0.42\end{pmatrix}$...
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Does Covariance equal 0
Let $U,V$ be a bivariate random variable with a continuous distribution and $f_{U,V}$ is the joint density of $(U,V)$. Suppose that $f_{U,V}(−u,v)=f_{U,V}(u,v)$ for all $u,v∈\mathbb{R}$, then $cov(U,V)...
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Expectation of $|x^{2} - y^{2}|$ when $x$ and $y$ are bivariate normal
I was asked the following question by my professor. Suppose $x$ and $y$ are jointly normal variables. We further suppose the marginal distribution of $x$ and $y$ are $N(0, 1)$ and $N(0, 2)$ ...
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Finding max of $var(a^TY)$.
Let $X=(X_1,X_2,X_3)^T$ be a multivariate random variable with the standard Gaussian distribution on $\mathbb{R}^3$. Define the multivariate random variable $Y=(Y_1,Y_2,Y_3)^T$ by
$$\begin{pmatrix} ...
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1
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Deriving the joint probability distribution of a transformed rv
Suppose that A and B are 2 r.v with joint probability distribution given by $f_{AB}(a,b) = -\frac{1}{2}(ln(a)+ln(b))$, if $0 \lt a \lt 1$ and $0 \lt b \lt 1$, and 0 otherwise.
Define $X = A+B$ and $Y =...
user986741
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Finding the probability of a bivariate joint random variable
Let $A=(M,N)^T$ be a bivariate random variable with joint density defined by
$$f_{M,N}(m,n) = \frac{3}{2 \pi} \sqrt{m^2+n^2}$$ if $m^2+n^2<1$ and $0$ otherwise.
Let $B = (F,G)^T$ be given by
$$\...
user986741
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0
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76
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Absolute value of expected value
Given $(A_1,A_2)$ to be the bivariate random variable in $\mathbb{R}^2$ with mean $0$ and cov($A_1,A_2)$ $= -1.05$ and $Var(Z_1) = Var(Z_2) = 1.05$.
How do I find the expected value of $E(|A_1A_2|)$? ...
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Finding the expected value of a bivariate gaussian distribution
Suppose that $(A,B)$ have a standard Gaussian distribution on $\mathbb{R}^2$.
How do I find the expected value for $\mathbb{E}[max(3.9A+B,A+3.9B)]$?
I know that A and B follow the standard normal ...
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1
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Proof that $\mathbb{E}\bigl[1-\Phi(x_1 - \theta_1) \mid \theta_2 \leq x_2-σx_1+σθ_1\bigr]$ is increasing in $\sigma$
Numerically it appears that the following function is increasing (or at least non-decreasing) in $σ$,
$$
f(x_1,x_2;σ)=∫_{-∞}^{∞}∫_{-∞}^{x_2-σx_1+σθ_1}[1-Φ(x_1-θ_1)]φ(θ_1)φ(θ_2)dθ_2dθ_1
$$
where $\phi$ ...
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Finding $P(X \leq a | Y = k)$ where $X\sim\operatorname{Exp}(\lambda)$ and $Y = [X]$ where $[\cdot]$ rounds up to the nearest integer.
$Y = [X]$ where $[\cdot]$ rounds up to the nearest integer. It's given that $X \sim \operatorname{Exp}(\lambda)$. The first part asks me to show that $Y \sim \operatorname{Geo}(1-e^{-\lambda})$. The ...
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Guessing a function that maximises the variance?
I'm doing a question that discusses a six-sided fair die being rolled and then discussing the numbers that appear on the top and on the side facing you. The rest of the question discusses the ...
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Bivariate Normal Distribution and sub vector
Let $$(X,Y)\sim N(\mu^\rightarrow ,\Sigma)$$
we cannot assume anything about the dependancy between X and Y.
Can we assume the following? $$X\sim N(\mu_x,\sigma ^2_x)~,~Y\sim N(\mu_y, \sigma^2_y)$$...
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how to use correlation to calculate condition expectation
Suppose (X, Y) is Bivariate Normal, with X,Y ~ N(0, 1), and Corr(X, Y) = $\rho$ , please find E(Y|X) and E(X|Y).
I don't have a clue how to solve this, anyone could give me a pointer? Just for the ...
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